Since \(\widetilde{\mathcal {L}}\)  is a kernel operator with kernel \(k_w\in L^2(pw\otimes pw)\) , \(\widetilde{\mathcal {L}}\)  is a continuous Hilbert-Schmidt operator on \(L^2(pw)\)  by Theorem VI.23 in [1]}. Point (e) of Theorem VI.22 in [1]} ensures that every Hilbert-Schmidt operator is compact. As \(L^2(pw)=E^2(p,w)\)  by Proposition , we reap the first statement.
The first inequality is obvious since \(C(p,w)=\kappa _1\) .
Theorems VI.22 and VI.23 in [1]} tell us that \(\left\Vert k_w\right\Vert _{L^2(pw\otimes pw)}^2=\text{tr}(\widetilde{\mathcal {L}}^2)\)  where tr denotes the trace of an operator. By Theorem VI.18 in [1]}, the trace of \(\widetilde{\mathcal {L}}^2\)  is
\(\text{tr}(\widetilde{\mathcal {L}}^2)=\sum _{i\in \mathbb {N}}\mathbb {E}_p\left[e_i\widetilde{\mathcal {L}}^2 e_i\right]=\sum _{i\in \mathbb {N}}\mathbb {E}_p\left[\widetilde{\mathcal {L}}e_i\widetilde{\mathcal {L}}e_i\right]=\sum _{i\in \mathbb {N}}\kappa _i^2\) 
where we have used the fact that \(\widetilde{\mathcal {L}}\)  is self-adjoint and (A1)-(A2).
